\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 17, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/17\hfil Nonlocal BVPs for n-th Order ODEs]
{Nonlocal boundary-value problems for n-th order ordinary
differential equations by matching solutions}

\author[X. Liu\hfil EJDE-2011/17\hfilneg]
{Xueyan Liu}

\address{Xueyan (Sherry) Liu \newline
Department of Mathematics \\
Baylor University \\
Waco, TX 76798-7328, USA}
\email{Xueyan\_Liu@baylor.edu}

\thanks{Submitted June 26, 2010. Published February 3, 2011.}
\subjclass[2000]{34B15, 34B10}
\keywords{Boundary value problem; nonlocal; matching solutions}

\begin{abstract}
 We are concerned with the existence and uniqueness of solutions to
 nonlocal boundary-value problems on an interval $[a,c]$ for the
 differential equation $y^{(n)}=f(x,y,y',\dots,y^{(n-1)})$,
 where $n\geq 3$. We use the method of matching solutions, with some
 monotonicity conditions on $f$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks

\section{Introduction}

In this article, we are concerned with the existence and
uniqueness of solutions of boundary-value problems (BVP's)
 for the differential equation
\begin{gather}\label{e1}
y^{(n)}(x)=f(x,y(x),y'(x),\dots,y^{(n-1)}(x)),\quad
 n\geq 3,\; x\in[a,c],\\
\label{e2}
\begin{gathered}
y(a)-\sum_{i=1}^{s}\alpha_iy(\xi_i)=y_1,\quad
y^{(i)}(b)=y_{i+2},\quad  0\leq i\leq n-3,\\
\sum_{j=1}^{t}\beta_j y(\eta_j)-y(c)=y_n,
\end{gathered}
\end{gather}
where $a<\xi_1<\xi_2<\dots<\xi_s<b<\eta_1<\eta_2<\dots<\eta_t<c$,
$s,t\in {\mathbb{N}}$,  $\alpha_i> 0$ for $1\leq i\leq s$,
$\beta_j> 0$ for $1\leq j\leq t$, $\sum_{i=1}^{s}\alpha_i=1$,
$\sum_{j=1}^{t}\beta_j=1$, and $y_1,y_2,\dots,y_n\in \mathbb{R}$.

It is assumed throughout that $f:[a,c]\times \mathbb{R}^{n}\to
\mathbb{R}$ is continuous and that solutions for the initial value
problems (IVP's) for \eqref{e1} are unique and exist on $[a,c]$.
Moreover
$a<\xi_1<\xi_2<\dots<\xi_s<b<\eta_1<\eta_2<\dots<\eta_t<c$ are
fixed throughout.

Consider the following boundary conditions:
\begin{gather}\label{e3}
y(a)-\sum_{i=1}^{s}\alpha_iy(\xi_i)=y_1,\quad
 y^{(i)}(b)=y_{i+2},\; 0\leq i\leq n-3, \quad y^{(n-2)}(b)=m,\\
 \label{e4}
y(a)-\sum_{i=1}^{s}\alpha_iy(\xi_i)=y_1,\quad
 y^{(i)}(b)=y_{i+2},\; 0\leq i\leq n-3, \quad
y^{(n-1)}(b)=m,\\
\label{e5}
y^{(i)}=y_{i+2},\; 0\leq i\leq n-3,\quad
y^{(n-2)}(b)=m,\quad \sum_{j=1}^{t}\beta_jy(\eta_j)-y(c)=y_n,\\
\label{e6}
y^{(i)}=y_{i+2},\; 0\leq i\leq n-3,\quad
y^{(n-1)}(b)=m,\quad \sum_{j=1}^{t}\beta_jy(\eta_j)-y(c)=y_n,
\end{gather}
where $m\in \mathbb{R}$. We show that
\eqref{e1}-\eqref{e2} has a unique solution by matching
solutions of the BVP's \eqref{e1}-\eqref{e3} on $[a,b]$
and \eqref{e1}-\eqref{e5} on $[b,c]$, or \eqref{e1}-\eqref{e4}
 on $[a,b]$ and \eqref{e1}-\eqref{e6} on $[b,c]$.

The method of matching solutions was first used by Bailey
 et al. \cite{Bailey68}. They considered the solutions of
two-point boundary value problems for the second order
differential equation $y''=f(x,y,y')$ by matching solutions
of initial value problems. After that, in 1973,
Barr and Sherman \cite{Barr73} applied the solution matching
techniques to third order equations and generalized to equations
of arbitrary order. A  monotonicity condition on $f$ played an
important role. In 1981, Rao et al. \cite{Rao81} generalized
the monotonicity of $f$ of third order differential equations
and introduced an auxiliary monotone function $g$. In 1983,
Henderson \cite{Henderson83} generalized to $n$th order BVP's
and considered more general boundary conditions. Since then
there has been a lot of literature dealing with solutions
of third order BVP's or higher order BVP's by using matching
solutions; see
\cite{Barr74, Henderson09, Henderson01, Taunton93, Henderson05, Moorti78}, etc.

In this article, we consider the $n$-th order BVP's with nonlocal
boundary conditions \eqref{e1}-\eqref{e2} and use a weaker
condition on the auxiliary function $g$.
In Section 2, we give some preliminary results, and in Section 3,
we prove the existence and uniqueness of solutions of
\eqref{e1}-\eqref{e2}. In Section 4, we generalize our
results to BVP's with more general boundary conditions:
\begin{equation} \label{e8}
\begin{gathered}
y^{(\tau)}(a) -\sum_{i=1}^{s}\alpha_iy^{(\tau)}
(\xi_i)=y_1,\quad y^{(i)}(b)=y_{i+2},\;
0\leq i\leq n-3,\\
\sum_{j=1}^{t}\beta_j y^{(\sigma)}(\eta_j)-y^{(\sigma)}(c)=y_n,
\end{gathered}
\end{equation}
with $\tau, \sigma\in \{0,1,\dots,n-3\}$ fixed.

We assume there is a continuous function
$g:[a,c]\times \mathbb{R}^{n}\to \mathbb{R}$ and
that $f$ and $g$ satisfy the following conditions:
\begin{itemize}

\item[(A)] For  $u,v\in \mathbb{R}$,
$f(x,v_0,v_1,\dots,v_{n-2},v)-f(x,u_0,u_1,\dots,
u_{n-2},u)>g(x,v_0-u_0,v_1-u_1,\dots,v_{n-2}-u_{n-2},v-u)$ when
$x\in (a,b]$, $(-1)^{n-i}v_i\geq (-1)^{n-i}u_i$, $0\leq i\leq
n-3$, and $v_{n-2}>u_{n-2}$; or when $x\in [b,c)$, $v_i\geq u_i$,
$0\leq i\leq n-3$, and  $v_{n-2}>u_{n-2}$.

\item[(B)] There exists $\delta_1>0$, such that for all
$0<\delta<\delta_1$, the IVP
\begin{gather}\label{e9}
z^{(n)}=g(x,z,z',\dots, z^{(n-1)}),\\ \label{e10}
z^{(i)}(b)=0,\quad 0\leq i\leq n-1,\quad i\neq n-2,\quad
z^{(n-2)}(b)=\delta
\end{gather}
has a solution $z$ on $[a,c]$ such that $z^{(n-2)}(x)\geq 0$
on $[a,c]$.

\item[(C)] There exists $\delta_2>0$, such that for all
$ 0<\delta<\delta_2$, the IVP
\begin{gather}\label{e11}
 z^{(n)}=g(x,z,z',\dots, z^{(n-1)}),\\
\label{e12}
z^{(i)}(b)=0,\quad 0\leq i\leq n-2,\quad
z^{(n-1)}(b)=\delta, (-\delta)
\end{gather}
has a solution $z$ on $[b,c]$ ($[a,b]$) such that
$z^{(n-2)}(x)\geq 0$ on $[b,c]$, ($z^{(n-2)}(x)\geq 0$ on $[a,b]$).

\item[(D)] For each $w\in \mathbb{R}$,
$g(x,v_0,v_1,\dots,v_{n-2}, w)\geq
g(x,u_0,u_1,\dots,u_{n-2}, w)$ when $x\in (a,b]$,
$(-1)^{n-i}(v_i-u_i)\geq 0$, $i=0,1,\dots,n-3$, and
$v_{n-2}>u_{n-2}\geq 0$, or when $x\in [b,c)$, $v_i\geq u_{i}$,
$i=0,1,\dots, n-3$, and $v_{n-2}>u_{n-2}\geq 0$.
\end{itemize}

\section{Preliminaries}

In this section, we give two lemmas which show the relationship
 between the value of the $n-2$nd order and the $n-1$st order of
two solutions of \eqref{e1} at $b$ that satisfy the boundary
conditions (2), respectively, on the interval $[a,b]$ and the
interval $[b,c]$. All of the results in Section 3 are based
on  two lemmas.
We basically prove the lemmas by using contradiction.

\begin{lemma}\label{lemma1}
Suppose $p$ and $q$ are solutions of \eqref{e1} on $[a,b]$ and
$w=p-q$ satisfies the following boundary conditions:
$$
w(a)-\sum_{i=1}^{s}\alpha_iw(\xi_i)=0,\quad
w^{(i)}(b)=0,\quad 0\leq i\leq n-3.
$$
Then, $w^{(n-2)}(b)=0$ if and only if $w^{(n-1)}(b)=0$.
 Also, $w^{(n-2)}(b)>0$ if and only if $w^{(n-1)}(b)>0$.
\end{lemma}

\begin{proof}
$(\Rightarrow)$ The necessity of the first part.
Suppose $w^{(n-2)}(b)=0$ and  $w^{(n-1)}(b)\neq 0$.
Without loss of generosity, we assume $w^{(n-1)}(b)<0$.

Since $0=w(a)-\sum_{i=1}^{s}\alpha_iw(\xi_i)
=\sum_{i=1}^{s}\alpha_i(w(a)-w(\xi_i))$ and $\alpha_i> 0$,
for some $i_1$, $w(a)\geq w(\xi_{i_1})$, and for some
$i_2$, $w(a)\leq w(\xi_{i_2})$. Hence, there exists
$r_1\in (a,b)$ such that $w'(r_1)=0$ and $(-1)^{n-1}w'(x)>0$
on $(r_1,b)$.

By repeated applications of Rolle's Theorem, there exists $r_2 \in
(r_1,b)$ such that $w^{(n-2)}(r_2)=0$ and $w^{(n-2)}(x)>0$, for
$x\in (r_2,b)$. Hence, $(-1)^{n-j}w^{(j)}(x)> 0$, for
$j=0,1,\dots,n-2$, on $(r_2,b)$.

Let $\delta \in \mathbb{R}$ with
$0<\delta<\min\{\delta_2, -w^{(n-1)}(b)\}$. Then, by Condition (C),
we have a solution $z$ of \eqref{e11}-\eqref{e12} on $[a,b]$,
such that  $ z^{(i)}(b)=0$, $0\leq i\leq n-2$,
$z^{(n-1)}(b)=-\delta$, and $z^{(n-2)}(x)\geq 0$ on $[a,b]$.

Let $h=w-z$. Then, we have
\begin{gather*}
 h^{(n)}=f(x,p,p',\dots, p^{(n-1)})-f(x,q,q',\dots, q^{(n-1)})-g(x,z,z',\dots,z^{(n-1)}),\\
 h^{(i)}(b)=0, \; 0\leq i\leq n-2, \quad
h^{(n-1)}(b)=w^{(n-1)}(b)-z^{(n-1)}(b)<0.
\end{gather*}

Notice $h^{(n-2)}(r_2)=w^{(n-2)}(r_2)-z^{(n-2)}(r_2)\leq 0$,
$h^{(n-2)}(b)=0$ and $h^{(n-1)}(b)<0$. So there exists
$r_3\in [r_2, b)$ such that $h^{(n-2)}(r_3)=0$ and
$h^{(n-2)}(x)>0$ for $x\in (r_3,b)$. Then, it follows that
$(-1)^{n-j}h^{(j)}(x)>0$ on
$(r_3,b)$, for $j=0,1,\dots, n-2$. Therefore, by Rolle's Theorem,
there is $r_4\in (r_3, b)$ such that $h^{(n-1)}(r_4)=0$. Since
$h^{(n-1)}(b)<0$, there is $r_5\in [r_4, b)$ such that
$h^{(n-1)}(r_5)=0$ and $h^{(n-1)}(x)<0$ for $x\in (r_5, b)$. Then,
\[
h^{(n)}(r_5)=\lim_{x\to r_5^+}
\frac{h^{(n-1)}(x)-h^{(n-1)}(r_5)}{x-r_5} \leq 0,
\]
whereas by Conditions (A) and (D),
(note that $[r_5,b)\subset (r_3,b)\subset (r_2,b)$),
\begin{align*}
h^{(n)}(r_5)
&= f(r_5,p,p',\dots, p^{(n-1)})-f(r_5,q,q',\dots, q^{(n-1)})
 -g(r_5,z,z',\dots,z^{(n-1)})\\
&> g(r_5,w,w',\dots,w^{(n-1)})-g(r_5,z,z',\dots,z^{(n-1)})
\geq  0,
\end{align*}
which is a contradiction. Therefore, $w^{(n-1)}(b)= 0$.


$(\Leftarrow)$ The sufficiency of the first part.
Suppose $w^{(n-1)}(b)=0$ and  $w^{(n-2)}(b)\neq 0$.
Without loss of generality, we assume $w^{(n-2)}(b)>0$.

Since $0=w(a)-\sum_{i=1}^{s}\alpha_iw(\xi_i)
=\sum_{i=1}^{s}\alpha_i(w(a)-w(\xi_i))$ and $\alpha_i> 0$,
there exists $r_1\in (a,b)$ such that $w'(r_1)=0$, and
$(-1)^{n-1}w'(x)>0$ on $(r_1,b)$.


By repeated applications of Rolle's Theorem, there exists $r_2 \in
(r_1,b)$ such that $w^{(n-2)}(r_2)=0$ and $w^{(n-2)}(x)>0$ for
$x\in (r_2,b)$. Hence, $(-1)^{n-j}w^{(j)}(x)> 0$, for
$j=0,1,\dots,n-2$, on $(r_2,b)$.

Now let $0<\delta<\min\{\delta_1,w^{(n-2)}(b)\}$, and let $z$
be a solution of \eqref{e9}-\eqref{e10} satisfying Condition
(B) and $z^{(n-2)}(x)\geq 0$ on $[a,b]$.
Let $h=w-z$. Then,
\begin{gather*}
 h^{(n)}=f(x,p,p',\dots, p^{(n-1)})-f(x,q,q',\dots, q^{(n-1)})-g(x,z,z',\dots,z^{(n-1)}),\\
 h^{(i)}(b)=0, \; 0\leq i\leq n-1,\; i\neq n-2, \quad
h^{(n-2)}(b)=w^{(n-2)}(b)-z^{(n-2)}(b)>0.
\end{gather*}

Note that $h^{(n-2)}(r_2)=w^{(n-2)}(r_2)-z^{(n-2)}(r_2)\leq 0$.
Hence, there is $r_3\in [r_2,b)$ such that $h^{(n-2)}(r_3)=0$,
$h^{(n-2)}(x)>0$ on $(r_3,b)$. By Rolle's Theorem, there is
$r_4\in (r_3,b)$ such that $h^{(n-1)}(r_4)>0$ and
$(-1)^{n-j}h^{(j)}(x)>0$ on $(r_4,b)$, for $j=0,1,\dots, n-2$.

By Conditions (A) and (D),
\begin{align*}
h^{(n)}(b)
&= f(b,p,p',\dots, p^{(n-1)})-f(b,q,q',\dots, q^{(n-1)})
 -g(b,z,z',\dots,z^{(n-1)})\\
&> g(b,w,w',\dots,w^{(n-1)})-g(b,z,z',\dots,z^{(n-1)})
\geq  0.
\end{align*}

Together with $h^{(n-1)}(b)=0$, we have that $h^{(n-1)}(x)<0$ on
a left neighborhood of $b$. Since $h^{(n-1)}(r_4)>0$, there
is $r_5\in (r_4,b)$ such that $h^{(n-1)}(r_5)=0$ and $h^{(n-1)}(x)<0$
on $(r_5,b)$. Hence, $h^{(n)}(r_5)\leq 0$.

However, (note that $[r_5,b)\subset(r_4,b)\subset(r_2,b)$),
\begin{align*}
h^{(n)}(r_5)
&= f(r_5,p,p',\dots, p^{(n-1)})-f(r_5,q,q',\dots, q^{(n-1)})
 -g(r_5,z,z',\dots,z^{(n-1)})\\
&> g(r_5,w,w',\dots,w^{(n-1)})-g(r_5,z,z',\dots,z^{(n-1)})
\geq  0,
\end{align*}
which is a contradiction. Hence, our assumption is false.

$(\Rightarrow)$ The necessity of the second part.
Assume $w^{(n-1)}(b)<0$ and $w^{(n-2)}(b)>0$. Similar to the proof
of the first part, we have $r_1\in(a,b)$ such that
$w^{(n-2)}(r_1)=0$ and $w^{(n-2)}(x)>0$, for $x\in (r_1,b)$ and
$(-1)^{n-j}w^{(j)}(x)>0$ on $(r_1,b)$, for $j=0,1,\dots,n-2$.

Now let $0<\delta<\min\{\delta_1,w^{(n-2)}(b)\}$, and let
 $z$ be a solution of \eqref{e9}-\eqref{e10} satisfying
Condition (B) and $z^{(n-2)}(x)\geq 0$ on $[a,b]$.
Let $h=w-z$. Then,
\begin{gather*}
 h^{(n)}=f(x,p,p',\dots, p^{(n-1)})-f(x,q,q',\dots, q^{(n-1)})-g(x,z,z',\dots,z^{(n-1)}),\\
 h^{(i)}(b)=0,\; 0\leq i\leq n-3, \quad
h^{(n-2)}(b)=w^{(n-2)}(b)-z^{(n-2)}(b)>0.
\end{gather*}

Note that $h^{(n-1)}(b)=w^{(n-1)}(b)-z^{(n-1)}(b)=w^{(n-1)}(b)<0$,
$h^{(n-2)}(b)>0$ and
$h^{(n-2)}(r_1)=w^{(n-2)}(r_1)-z^{(n-2)}(r_1)=-z^{(n-2)}(r_1)\leq
0$. So  there exists $r_2\in [r_1,b)$ such that
$h^{(n-2)}(r_2)=0$, $h^{(n-2)}(x)>0$, for $x\in (r_2,b)$. It
follows that $(-1)^{n-j}h^{(j)}(x)>0$ on $(r_2,b)$, for
$j=0,1,\dots, n-2$.

By Rolle's Theorem and $h^{(n-1)}(b)<0$, there is
$r_3\in(r_2,b)$ such that $h^{(n-1)}(r_3)=0$ and
$h^{(n-1)}(x)< 0$ on $(r_3,b)$.
Therefore, $h^{(n)}(r_3)\leq 0$, whereas
by Conditions (A) and (D),
(note that $[r_3,b)\subset (r_2,b)\subset (r_1,b)$),
\begin{align*}
h^{(n)}(r_3)
&= f(r_3,p,p',\dots, p^{(n-1)})-f(r_3,q,q',\dots, q^{(n-1)})
 -g(r_3,z,z',\dots,z^{(n-1)})\\
&> g(r_3,w,w',\dots,w^{(n-1)})-g(r_3,z,z',\dots,z^{(n-1)})
\geq  0,
\end{align*}
which is a contradiction.

$(\Leftarrow)$ The sufficiency of the second part.
We assume that $w^{(n-1)}(b)>0$ and  $w^{(n-2)}(b)<0$.
Then, we get the same situation as the proof of necessity
with opposite signs of $w^{(n-1)}(b)$ and $w^{(n-2)}(b)$,
which also implies a contradiction. Hence, the sufficiency is true.
\end{proof}

\begin{lemma}\label{lemma2}
Suppose $p$ and $q$ are solutions of \eqref{e1} on $[b,c]$ and
$w=p-q$ satisfies the following boundary conditions:
$$
w^{(i)}(b)=0,\; 0\leq i\leq n-3,\quad
 \sum_{j=1}^{t}\beta_jw(\eta_j)-w(c)=0.
$$
Then, $w^{(n-2)}(b)=0$ if and only if $w^{(n-1)}(b)=0$.
 Also, $w^{(n-2)}(b)>0$ if and only if $w^{(n-1)}(b)<0$.
\end{lemma}

\begin{proof}
$(\Rightarrow)$ The necessity of the first part.
Assume $w^{(n-2)}(b)=0$ and for contradiction, without loss
of generality, we assume $w^{(n-1)}(b)>0$.

By $\sum_{j=1}^{t}\beta_jw(\eta_j)-w(c)=0$, there exists
$r_1\in (b,c)$ such that $w'(r_1)=0$. By repeated applications
of Rolle's Theorem, there exists $r_2\in (b,r_1)$ such that
$w^{(n-2)}(r_2)=0$ and $w^{(n-2)}(x)>0$ on $(b,r_2)$. It follows
that $w^{(j)}(x)>0$ on $(b,r_2)$, for $j=0,1,\dots,n-2$.

Let $0<\delta<\min\{\delta_2, w^{(n-1)}(b)\}$. Then, by Condition
(C), we have a solution $z$ of \eqref{e11}-\eqref{e12} on $[b,c]$
such that $ z^{(i)}(b)=0$, $0\leq i\leq n-2$, $z^{(n-1)}(b)=\delta$,
and $z^{(n-2)}(x)\geq 0$ on $[b,c]$. Then, $z^{(j)}(x)\geq 0$, for
$j=0,1,\dots, n-2$, on $[b,c]$.

Let $h=w-z$. Then,
\begin{gather*}
 h^{(n)}=f(x,p,p',\dots, p^{(n-1)})-f(x,q,q',\dots,
 q^{(n-1)})-g(x,z,z',\dots,z^{(n-1)}),\\
 h^{(i)}(b)=0, ~0\leq i\leq n-2, ~~h^{(n-1)}(b)=w^{(n-1)}(b)
 -z^{(n-1)}(b)>0.
\end{gather*}

Note that $h^{(n-2)}(r_2)=w^{(n-2)}(r_2)-z^{(n-2)}(r_2)\leq 0$.
Hence, there is $r_3\in (b,r_2]$ such that $h^{(n-2)}(r_3)=0$,
$h^{(n-2)}(x)>0$ on $(b,r_3)$, and hence, $h^{(j)}(x)> 0$, for
$j=0,1,\dots, n-2$ on $(b,r_3)$. By $h^{(n-2)}(b)=0$, Rolle's
Theorem, and $h^{(n-1)}(b)>0$, there exists $r_4\in (b,r_3)$ such
that $h^{(n-1)}(r_4)=0$ and $h^{(n-1)}(x)>0$ on $(b,r_4)$. Hence,
$h^{(n)}(r_4)\leq0$, but by Conditions (A) and (D) and
$(b,r_4]\subset(b,r_3)\subset(b,r_2)$, we have
\begin{align*}
h^{(n)}(r_4)
&= f(r_4,p,p',\dots, p^{(n-1)})-f(r_4,q,q',\dots, q^{(n-1)})
 -g(r_4,z,z',\dots,z^{(n-1)})\\
&> g(r_4,w,w',\dots,w^{(n-1)})-g(r_4,z,z',\dots,z^{(n-1)})
\geq  0,
\end{align*}
which is a contradiction.

$(\Leftarrow)$ The sufficiency of the first part.
Suppose $w^{(n-1)}(b)=0$ and $w^{(n-2)}(b)>0$.
Similar to the above, we have $r_1\in (b,c)$ such that
$w^{(n-2)}(r_1)=0$ and $w^{(j)}(x)>0$ on $(b,r_1)$ for
$j=0,1,\dots,n-2$.

Let $0<\delta<\min\{\delta_1, w^{(n-2)}(b)\}$. Then, by Condition
(B), we have a solution $z$ of \eqref{e9}-\eqref{e10} on $[b,c]$
such that $ z^{(i)}(b)=0$, $0\leq i\leq n-1$,
$i\neq n-2$, $z^{(n-2)}(b)=\delta$, and $z^{(n-2)}(x)\geq 0$ on $[b,c]$.
Then, $z^{(j)}(x)\geq 0$, for $j=0,1,\dots, n-2$, on $[b,c]$.

Let $h=w-z$. Then,
\begin{gather*}
 h^{(n)}=f(x,p,p',\dots, p^{(n-1)})-f(x,q,q',\dots,
 q^{(n-1)})-g(x,z,z',\dots,z^{(n-1)})\\
 h^{(i)}(b)=0, \; 0\leq i\leq n-1,\; i\neq n-1,\quad
 h^{(n-2)}(b)=w^{(n-2)}(b)-z^{(n-2)}(b)>0.
\end{gather*}

Note that
$h^{(n-2)}(r_1)=w^{(n-2)}(r_1)-z^{(n-2)}(r_1)=-z^{(n-2)}(r_1)\leq
0$. So there is $r_2\in(b,r_1]$ such that $h^{(n-2)}(r_2)= 0$ and
$h^{(n-2)}(x)> 0$, for $x\in (b,r_2)$, and $h^{(j)}(x)> 0$ on
$(b,r_2)$, for $j=0,1,\dots,n-2$. By Rolle's Theorem, there is
$r_3\in (b,r_2)$ such that $h^{(n-1)}(r_3)<0$.

Note that
\begin{align*}
h^{(n)}(b)
&= f(b,p,p',\dots, p^{(n-1)})-f(b,q,q',\dots, q^{(n-1)})
 -g(b,z,z',\dots,z^{(n-1)})\\
&>  g(b,w,w',\dots,w^{(n-1)})-g(b,z,z',\dots,z^{(n-1)})
\geq  0.
\end{align*}
Hence, there is $r_4\in(b,r_3)$ such that $h^{(n-1)}(r_4)=0$ and
$h^{(n-1)}(x)> 0$ on $(b,r_4)$,  which implies $h^{(n)}(r_4) \leq 0$.
But by $(b,r_4]\subset(b,r_2)\subset(b,r_1)$ and Conditions (A) and (D), we have that
\begin{align*}
h^{(n)}(r_4)
&= f(r_4,p,p',\dots, p^{(n-1)})-f(r_4,q,q',\dots, q^{(n-1)})
 -g(r_4,z,z',\dots,z^{(n-1)})\\
&> g(r_4,w,w',\dots,w^{(n-1)})-g(r_4,z,z',\dots,z^{(n-1)})
\geq  0,
\end{align*}
which is a contradiction.

$(\Rightarrow)$ The necessity of the second part.
Suppose $w^{(n-2)}(b)>0$  and  $w^{(n-1)}(b)>0$. Similar to
the proof of the necessity of the first part, we also can get
a contradiction. Hence, we omit the proof. Therefore,
if $w^{(n-2)}(b)>0$, then  $w^{(n-1)}(b)<0$.

$(\Leftarrow)$ The sufficiency of the second part.
Suppose $w^{(n-1)}(b)<0$. If $w^{(n-2)}(b)<0$, then similar
to the proof of necessity, we can get $w^{(n-1)}(b)>0$,
which is a contradiction. Hence, the sufficiency is also true.
\end{proof}


\section{Existence and uniqueness of solutions to \eqref{e1}-\eqref{e2}}

Before discussing existence and uniqueness for \eqref{e1}-\eqref{e2},
we consider the uniqueness of solutions to each of the BVP's
for \eqref{e1} satisfying any of \eqref{e3}, \eqref{e4}, \eqref{e5},
 or \eqref{e6}.

\begin{lemma} \label{lemma3}
Let $y_1,y_2,\dots,y_n\in \mathbb{R}$ be given and assume
Conditions {\rm (A)--(D)} are satisfied. Then, given $m\in \mathbb{R}$,
each of the BVP's for \eqref{e1} satisfying any of conditions
\eqref{e3}, \eqref{e4}, \eqref{e5}, or \eqref{e6} has at most one
solution.
\end{lemma}

\begin{proof}
The case of \eqref{e1}-\eqref{e3}:
Suppose there are two distinct solutions $p(x)$ and $q(x)$ for
some $m\in \mathbb{R}$. Let $w=p-q$. Then, $w$ satisfies
\begin{gather*}
w^{(n)}=f(x,p,p',\dots, p^{(n-1)})-f(x,q,q',\dots, q^{(n-1)}),\\
w(a)-\sum_{i=1}^{s}\alpha_iw(\xi_i)=0, \quad
w^{(i)}(b)=0, \quad 0\leq i\leq n-2.
\end{gather*}
By Lemma \ref{lemma1}, we can get that $w^{(n-1)}(b)=0$.
Then, by the uniqueness of solutions of IVP's for \eqref{e1},
we can conclude that $p\equiv q$ on $[a,b]$.
Hence, \eqref{e1}-\eqref{e3} has at most one solution on $[a,b]$.

{The other cases:}
By using similar ideas and Lemma \ref{lemma1} and
Lemma \ref{lemma2}, we resolve the other cases.
\end{proof}


\begin{lemma}\label{lemma4}
Let $y_1,y_2,\dots,y_n\in \mathbb{R}$ be given. Assume Conditions
(A)-(D) are satisfied. Then, the BVP \eqref{e1}-\eqref{e2} has at
most one solution.
\end{lemma}

\begin{proof}
We argue by contradiction. Suppose for some values
$y_1,y_2,\dots,y_n\in \mathbb{R}$, there exist distinct solutions
$p$ and $q$ of \eqref{e1}-\eqref{e2}. Also, let $w=p-q$. Then,
from Lemma \ref{lemma1} and Lemma \ref{lemma2}, we get
$w^{(n-2)}(b)\neq 0$, $w^{(n-1)}(b)\neq 0$.

Without loss of generality, we suppose  $w^{(n-2)}(b)>0$.
Then, by Lemma \ref{lemma1}, $w^{(n-1)}(b)> 0$.
But by Lemma \ref{lemma2}, $w^{(n-1)}(b)<0$. This is a contradiction.
Hence, $p\equiv q$ on $[a,c]$.
\end{proof}


Next, we show that solutions of \eqref{e1} satisfying each
of \eqref{e3}, \eqref{e4}, \eqref{e5}, or \eqref{e6}
respectively are monotone functions of $m$ at $b$.
For notation purposes, given any $m\in \mathbb{R}$,
let $\alpha(x,m)$, $u(x,m)$, $\beta(x,m)$, $v(x,m)$ denote
the solutions, when they exist, of the boundary value problems
of \eqref{e1} satisfying \eqref{e3}, \eqref{e4}, \eqref{e5},
or \eqref{e6}, respectively.


\begin{lemma}\label{lemma5}
Suppose that {\rm (A)--(D)} are satisfied and that for each
$m\in \mathbb{R}$, there exist solutions of \eqref{e1}
 satisfying each of the conditions \eqref{e3}, \eqref{e4}, \eqref{e5},
 \eqref{e6}, respectively. Then, $\alpha^{(n-1)}(b,m)$ and
$\beta^{(n-1)}(b,m)$ are, respectively, strictly increasing and
strictly decreasing functions of $m$ with ranges all of $\mathbb{R}$.
\end{lemma}

\begin{proof}
The proof of $\{\alpha^{(n-1)}(b,m)|m\in \mathbb{R}\}=\mathbb{R}$
is the same as that in \cite[Theorem 2.4]{Henderson83}.
We omit it here.
\end{proof}

Similarly, we obtain monotonicity conditions on $u^{(n-2)}(b,m)$
and $v^{(n-2)}(b,m)$.

\begin{lemma}\label{lemma6}
Under the assumption of  Lemma \ref{lemma5}, the functions
$u^{(n-2)}(b,m)$ and $v^{(n-2)}(b,m)$ are, respectively,
 strictly increasing and strictly decreasing functions of $m$,
with ranges all $\mathbb{R}$.
\end{lemma}

Finally, we arrive at our existence result for \eqref{e1}-\eqref{e2},
which is obtained by solution matching.

\begin{theorem}
Assume the hypotheses of Lemma \ref{lemma5}. Then,
\eqref{e1}-\eqref{e2} has a unique solution.
\end{theorem}


\begin{proof}
We prove the existence from either Lemma \ref{lemma5} or
Lemma \ref{lemma6}. Making use of Lemma \ref{lemma6}, there
exists a unique $m_0\in \mathbb{R}$ such that
$u^{(n-2)}(b,m_0)=v^{(n-2)}(b,m_0)$. Then,
$$
y(x)=\begin{cases}
u(x,m_0), &  a\leq x\leq b,\\
v(x,m_0), &  b\leq x\leq c,
\end{cases}
$$
is a solution of \eqref{e1}-\eqref{e2} and by Lemma \ref{lemma4},
$y(x)$ is the unique solution.
\end{proof}

\section{Existence and uniqueness of solutions to
\eqref{e1}-\eqref{e8}}

We can obtain analogous results to those of Section 3 for
\eqref{e1}-\eqref{e8} with $\tau, \sigma\in \{0,1,\dots,n-3\}$
fixed. We obtain solutions to \eqref{e1}-\eqref{e8} by matching
solutions satisfying the following types of boundary conditions:
\begin{gather}\label{e13}
y^{(\tau)}(a)-\sum_{i=1}^{s}\alpha_iy^{(\tau)}(\xi_i)=y_1, \quad
 y^{(i)}(b)=y_{i+2},\; 0\leq i\leq n-3, \quad
 y^{(n-2)}(b)=m, \\
\label{e14}
y^{(\tau)}(a)-\sum_{i=1}^{s}\alpha_iy^{(\tau)}(\xi_i)=y_1,\quad
 y^{(i)}(b)=y_{i+2},\; 0\leq i\leq n-3, \quad
 y^{(n-1)}(b)=m, \\
\label{e15}
y^{(i)}(b)=y_{i+2},\; 0\leq i\leq n-3, \quad
 y^{(n-2)}(b)=m, \quad
 \sum_{j=1}^{t}\beta_j y^{(\sigma)}(\eta_j)-y^{(\sigma)}(c)=y_n, \\
\label{e16}
y^{(i)}(b)=y_{i+2},\; 0\leq i\leq n-3, \quad
y^{(n-1)}(b)=m, \quad
\sum_{j=1}^{t}\beta_j y^{(\sigma)}(\eta_j)-y^{(\sigma)}(c)=y_n,
\end{gather}
where $m\in \mathbb{R}$,
$a<\xi_1<\xi_2<\dots<\xi_s<b<\eta_1<\eta_2<\dots<\eta_t<c$,
$s,t\in \mathbb{N}$,  $\alpha_i> 0$ for $1\leq i\leq s$,
$\beta_j> 0$ for $1\leq j\leq t$, $\sum_{i=1}^{s}\alpha_i=1$,
$\sum_{j=1}^{t}\beta_j=1$ and $y_1,y_2,\dots,y_n\in \mathbb{R}$.

We omit the proofs of the following results since they are
essentially the same as those presented in Section 2 with
only small modifications.

\begin{lemma} \label{lemma7}
Let $y_1,y_2,\dots,y_n\in \mathbb{R}$ be given and assume
conditions {\rm (A)--(D)} are satisfied. Then, given $m\in \mathbb{R}$,
each of the BVP's for \eqref{e1} satisfying any of conditions
\eqref{e13}, \eqref{e14}, \eqref{e15}, or \eqref{e16} has at most
one solution.
\end{lemma}

\begin{lemma} \label{lemma8}
Let $y_1,y_2,\dots,y_n\in \mathbb{R}$ be given and assume
conditions (A)-(D) are satisfied. Then
\eqref{e1}-\eqref{e8} has at most one solution.
\end{lemma}

Now, given any $m\in \mathbb{R}$, let $\theta(x,m)$, $l(x,m)$,
$\vartheta(x,m)$, $o(x,m)$ denote the solutions, when they exist,
of the boundary value problems of \eqref{e1} satisfying
\eqref{e13}, \eqref{e14}, \eqref{e15}, \eqref{e16}, respectively.


\begin{lemma} \label{lemma9}
Suppose that {\rm (A)--(D)} are satisfied and that for each
$m\in \mathbb{R}$, there exist solutions of \eqref{e1}
satisfying each of the conditions \eqref{e13}, \eqref{e14},
 \eqref{e15}, \eqref{e16}. Then, $\theta^{(n-1)}(b,m)$ and
$\vartheta^{(n-1)}(b,m)$ are respectively strictly increasing and
strictly decreasing functions of $m$ with ranges all of $\mathbb{R}$.
 Also, $l^{(n-2)}(b,m)$ and $o^{(n-2)}(b,m)$ are respectively
strictly increasing and strictly decreasing functions of $m$
with ranges all of $\mathbb{R}$.
\end{lemma}

\begin{theorem}
Assume the hypotheses of Lemma \ref{lemma9}. Then
 \eqref{e1}-\eqref{e8} has a unique solution.
\end{theorem}

\subsection*{Acknowledgements}
I am  grateful to Prof. Johnny Henderson who is my mentor
and has given me a lot of encouragement, professional advice, and
great comments about the original draft.


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\end{document}